Download Numerical modeling of turbulence mixed convection heat transfer in

Int. Jnl. of Multiphysics Volume 5 · Number 4 · 2011
307
Numerical modeling of turbulence mixed
convection heat transfer in air filled enclosures by
finite volume method
Mohammad Reza Safaei1*, Marjan goodarzi1 and
Mohammadali Mohammadi2
1Mashhad Branch, Islamic Azad University, Mashhad, Iran
2University Putra Malaysia, Serdang, Malaysia
ABSTRACT
In the present study, first the turbulent natural convection and then laminar
mixed convection of air flow was solved in a room and the calculated
outcomes are compared with results of other scientists and after showing
validation of calculations, aforementioned flow is solved as a turbulent
mixed convection flow, using the valid turbulence models Standard k−ε,
RNG k−ε and RSM.
To solve governing differential equations for this flow, finite volume
method was used. This method is a specific case of residual weighting
method. The results show that at high Richardson Numbers, the flow is
rather stationary at the center of the enclosure. Moreover, it is distinguished
that when Richardson Number increases the maximum of local Nusselt
decreases. Therefore, it can be said that less number of Richardson
Number, more rate of heat transfer.
Keywords: Mixed Convection Heat Transfer, Turbulence Models, Nusselt
Number, Turbulent Kinetic Energy, Reynolds Stress.
1. INTRODUCTION
Mixed convection heat transfer is a phenomenon in which both natural and forced
convections happen. Mixed convection heat transfer takes place either when buoyancy effect
matters in a forced flow or when there are sizable effects of forced flow in a buoyancy flow.
Dimensionless numbers to determine this type of flow are as follows: Grashof Number
(Gr = g.β.∆T.L3/v 2), Reynolds Number (Re = ρ.v.l/µ), Rayleigh Number (Ra = Gr.Pr),
Prandtl Number (Pr = Cp.µ/K), and Richardson Number (Ri). If you devide natural
convection effect by forced convection effect, it yields to Richardson number and it is written
as such: Ri = Gr/Re2. When it comes to limits and we have Ri→0 or Ri→∞, forced
convection and natural convection become dominant heat transfers respectively [Bejan [1]].
Mixed convection heat transfer is a fundamentally significant heat transfer mechanism
that occurs in selection industrial and technological applications.
In the recent years, wide and practical usages of mixed convection heat transfer in areas
such as designing solar collectors, double-layer glasses, building insulations, cooling
electronic parts and have attracted many scientists in studying it.
*Corresponding author. Tel.: +989151022063; E-mail address: [email protected]
308
Numerical modeling of turbulence mixed convection heat transfer in air filled
enclosures by finite volume method
Fluid flow and heat transfer in rectangular or square cavities driven have been studied
extensively in the literature. A review shows that there are two kinds of studies:
One way includes the entry of hot (or cold) fluid from one side, passing isothermal walls,
and exit from the other side. In this case, we could evaluate and compare the forced
convection effect caused by the entry and exit of the fluid. Some scientists have applied
thermal flux on the way fluid passes through the channel and then, they studied the effects
of it. Among the studies, we can mention the ones done by Rahman et al. [2], Saha et al. [3]
and Saha et al. [4]. Another method to create mixed convections is to move enclosure walls
in presence of hot (cold) fluid inside the enclosure. This creates shear stresses and provides
thermal and hydrodynamic boundary layers in the fluid inside the enclosure, and eventually
creates forced convection in it. Numerous studies have been conducted in this field so far.
We can mention the study done by Ghasemi & Aminossadati [5] as an instance. They have
studied heat transfer of natural convection in an inclined square enclosure that had two
insulated vertical walls and two horizontal walls with different temperatures with using finite
volume method. They studied pure water and CuO-water with 0.01 ≤ Φ ≤ 0.04. They varied
Rayleigh number between 103 and 107 and inclination angle between 0 Degrees to 90
Degrees and studied the impact of these factors on heat transfer and fluid flow in the
enclosure. They found that in low Rayleigh numbers - which heat transfer, is in a conduction
way — flow pattern and temperature in the inclination angles range of 30 degrees to 90
degrees is similar. However, for Rayleigh numbers bigger than 105, temperature and flow
pattern is different in inclination angle = 0 degree from other inclination angles.
Basak et al. [6] Studied the mixed convection flow inside a square enclosure with left and
right cold walls, insulated moving upper wall, and fixed lower hot wall by using finite element
method. They suggested that by increasing Gr, when Pr and Re are fixed, recirculation power
would improve.
In year 2007, Khanafer et al. [7] numerically studied unsteady mixed convection of a
simple fluid in a sinusoidal lid-sliding cavity. Their study showed that Grashof and Reynolds
numbers had an undeniable impact on nature and structure of the flow.
It is undeniable that the advancement in different sciences in the last decade has resulted
in much subtle laboratory measuring tools and it is true that using of modern methods like
parallel processing has enabled us to efficiently use numerical analysis methods. Yet analysis
of turbulent flows inside the enclosure is still a challenging topic in fluid mechanics. That is
because in experimental situation, it is too difficult to reach ideal adiabatic wall condition. It
is at the same time difficult to measure low speeds in enclosure boundary layers through
using present sensors and probes. Even numerically, although numerical methods like DES,
LES, and DNS have been subject to dramatic advancements, it is still nearly impossible to
predict the stratification in the core of the enclosure. Non-linearity and coupling of the
predominant equations have contributed into making the calculations complicated and time
consuming. That is while in designing large enclosures, Rayleigh number is usually large,
and so the flow nature is turbulent [Goshayeshi and safaei [8]]. The complexity of
calculations in mixed convection has made scientists to just study natural convection, among
which we can mention the studies Bessaih and Kadja [9], Ampofo [10], Salat et al. [11],
Xaman et al. [12] and Aounallah et al. [13]. In 2000, Tian and Karayiannis [14] started an
experimental study in South Bank University that was followed by Ampofo and Karayiannis
[15]. Data in this work were experimental benchmark data of natural convection flow inside
a square enclosure, and were used for other studies. Whereas Peng and Davidson [16] studied
the mentioned flow by using LES, and Omri and Galanis [17] used the SST k-ω to study this
flow. Hsieh and Lien [18] used turbulence models of steady RANS like Low-Re k-ε, and
Int. Jnl. of Multiphysics Volume 5 · Number 4 · 2011
309
numerically analyzed the works done by Betts and Bokhari [19] and Tian and Karayiannis
[14]. In the present work, turbulent natural convection inside square and rectangular
enclosures is modeled first, and the results have been compared with the studies Betts and
Bokhari [19] and Tian and Karayiannis [14], Ampofo and Karayiannis [15], Peng and
Davidson [16], Omri and Galanis [17] and Hsieh and Lien [18]. After the calculations being
valid, first the Laminar mixed convection flow is solved in the square enclosure, in
comparison with Basak et al. [6] and eventually turbulence mixed convection in square
enclosure is modeled for the first time in all of the world, by using turbulence models like
Standard k-ε, RNG k-ε and RSM.
2. PROBLEM FORMULATION
For modeling the investigated flow, continuity, we solve momentum, energy, and turbulence
equations. The properties have been considered fixed of course. Density is calculated
vertically by using variable density parameter for ∆T >30¡C and Boussinesq approximation
for ∆T<30¡C. The governing equations are as follow:
Continuity equation:
∂u ∂v
+
=0
∂x ∂y
(1)
Momentum equations in X and Y Direction:
 ∂u  ∂
 ∂u ∂v 
∂u
∂u
∂u
1 ∂p ∂
+u +v
=−
+ (υ + υt )  2  + (υ + υt )  + 
∂t
∂x
∂y
ρ ∂x ∂x
 ∂x  ∂y
 ∂y ∂x 
(2)
 ∂v  ∂
 ∂v ∂u 
∂v
1 ∂p
∂v
∂v
∂
+u +v = −
+ g β (T − Tm ) + (υ + υt )  2  + (υ + υt )  +  (3)
∂t
ρ ∂x
∂x
∂y
∂y
 ∂y  ∂x
 ∂x ∂y 
Energy equation:
v  ∂T ∂  v
v  ∂T
∂T
∂T
∂T
∂  v
+u
+v
=  + t 
+  + t 
∂t
∂x
∂y ∂x  Pr σ T  ∂x ∂y  Pr σ T  ∂y
(4)
Now for k-ε model, we will have:
Turbulent kinetic energy transport equation:
υ  ∂k ∂ 
υ  ∂k
∂ 
∂k
∂k
∂k
+ υ + t 
= υ + t 
+u +v
+ Pk + Gk − ε
∂y ∂x 
∂x
∂t
σ k  ∂y
σ k  ∂x ∂y 
(5)
Dissipation of turbulent kinetic energy transport equation:
υ  ∂ε ∂ 
υ  ∂ε
∂ε
∂ε
∂ε
∂ 
ε
ε2
ε
+u +v
= υ + t 
+ υ + t 
+ C1 Pk + C2
+ C3 Gk − Rε
∂t
∂x
∂y ∂x 
σ ε  ∂x ∂y 
σ ε  ∂y
k
k
k
(6)
The eddy viscosity obtained from Prandtl- Kolomogorov relation:
υt = C µ f µ
k2
ε
(7)
310
Numerical modeling of turbulence mixed convection heat transfer in air filled
enclosures by finite volume method
The stress production term, Pk, is modeled by:
2
2
  ∂u  2
 ∂v   ∂u ∂v  
Pk = υt  2   + 2   +  +  
  ∂x 
 ∂x   ∂y ∂y  


The buoyancy term, Gk, is defined by:
υ ∂T
Gk = − g β t
σ t ∂y
(8)
(9)
We will also have the following for RNG k-ε:

η
Cµ ρη3  1 −  2
 η0  ε
Rε =
k
1 + βη3
(10)
That:
η=
Sk
ε
(11)
The main difference between standard k-ε and RNG k-ε methods is in the ε equation, such
that we can say the RNG k-ε model is very same to standard k-ε model, but analytical
formulas for turbulent Prandtl Numbers have been improved. This is while these values in
standard k-ε model are gained experimentally.
For RSM model, the turbulence equations are as follows:
Reynolds stress transport equations:
( ) ( )
∂ dijk
D
ui′u ′j =
+ Pij + Gij + φij − ε ij
Dt
∂xk
(12)
That:
( ) ( )
( )
∂ ui′u ′j
∂ ui′u ′j
D
ui′u ′j =
+ uk
:
Dt
∂t
∂xk
dijk = v
( ) − P′ (u′δ
∂ uu′ u ′j
∂xk
ρ
i jk
( )
)
+ u ′jδ ik − ui′u ′j u′k :

∂u j
∂u i 
Pij = −  ui′uk′
+ u ′j uk′
:
∂xk
∂xk 

( )
Advection (By Mean Flow)
Diffusion
Production (By Mean Strain)
Int. Jnl. of Multiphysics Volume 5 · Number 4 · 2011
)
(
Gij = ui′ f j′ + u ′j f i ′ :
φij =
P ′  ∂ui′ ∂u ′j  2 P ′
+
S :

=
ρ  ∂x j ∂xi 
ρ ij
ε ij = 2v
∂ui′ ∂u ′j
:
∂xk ∂xk
311
Production (By Body Force)
Pressure-Strain Correlation
Dissipation
Turbulent kinetic energy transport equation:
(k )
Dk ∂di
=
+ P( k ) + G ( k ) − ε
∂xi
Dt
(13)
Except the terms Convection and Production in Reynolds stress transport equation, all the
other terms have contributed in introducing a series of correlations, which have to be
identified according to some known and unknown quantities, so that the equation system can
be configured.
Diffusion term:
− ui′u ′j uk′ = CS
∂ui′u ′j
k
uk′ ul′
ε
∂xl
(14)
Redistribution term:
φij = φij(1) + φij( 2 ) + φij( w)
(15)
That:

2
ε
φij(1) = −C1  ui′u ′j − kδ ij 
3
k



1
φij( 2 ) = −C2  Pij − Pkk δ ij 
3




3
3
φij( w) =  φkl nk nlδ ij − φik n j nk − φ jk ni nk  ψ
2
2


ε
φij = −C1( w) ui′u ′j + C2( w)φij( 2 )
k
(16)
312
Numerical modeling of turbulence mixed convection heat transfer in air filled
enclosures by finite volume method
3
k2
ψ=
ε
C3 yn
yn is the distance from the wall. The role of terms Φij(2), Φij(1) is to return isotropy (or
terminating anisotropic flow with distributing kinetic energy of Reynolds huge stresses
among the stresses of smaller size). The terms Φij(1) and Φij(2) are called return to isotropy
and isotropization of production , respectively. The term Φij(w) is named as wall reflection
term .
For Dissipation term, we have:
2
ε ij = δ ij ε
(17)
3
The constants in the above relations have been presented in table 1 for RNG k-ε, table 2
for standard k-ε, and in table 3 for RSM models [Safaei [20]].
In order to solve differential equations that govern on the flow, we will use finite volume
method, which is explained with details by Patankar [21] and Goshayeshi, Safaei and
Maghmoomi [22]. This method is a specific case of residual weighting methods. In this
approach, the computational field is divided to some control volumes in a way that a control
volume surrounds each node and control volumes have no volumes in common. The
differential equation is then integrated on each control volume. Profiles in pieces which show
changes (of a certain quantity like temperature, velocity, etc.) among the nodes, are used to
calculate the integrals. The result is discretization equation, which includes quantities for a
group of nodes.
In this way, the answer is always probed on the grid-points and interpolation formulas
(piecewise profiles) are necessary to calculate integrals and used within our calculations.
When we get discretized equations, these assumption profiles can be forgotten and we have
complete freedom to adopt various assumptions for these profiles. The advantage of this
method is high accuracy and exact integral balances even in coarse grids [Patankar [21]].
Table 1 Coefficients for RNG k-ε turbulent model
Cµ
0.0845
σk
σε
C1
C2
η0
β
K
1
1.3
1.42
1.68
4.38
0.012
0.41
Table 2 Coefficients for Standard k-ε turbulent model
Cµ
0.0845
σk
σε
C1
C2
1
1.3
1.42
1.68
Table 3 Coefficients for RSM turbulent model
(w
c2 )
(w
c1 )
0.3
0.5
Cs
0.22
C1
1.8
C2
0.6
C3
2.5
Int. Jnl. of Multiphysics Volume 5 · Number 4 · 2011
313
3. RESULTS
3.1. MESH INDEPENDENCY
Meshes that designed to cover control volumes, are square meshes provided on physical
domain with different distances in order to reach independence. The mentioned meshindependence for each turbulence model and any different Ri has been separately calculated.
Table 4 shows some meshes used in this study.
3.2. EVALUATION OF TURBULENT NATURAL CONVECTION FLOW INSIDE
THE ENCLOSURE
In order to demonstrate the accuracy of calculations, first of all, natural convection heat
transfer is solved in rectangular and square enclosures studied by other scientists like Betts
and Bokhari [19] and Tian and Karayiannis [14], Ampofo and Karayiannis [15], Peng and
Davidson [16], Omri and Galanis [17] and Hsieh and Lien [18].
After validating the results (see figures 1 and 2) with other scientists conclusions, it can
be concluded that the procedure of solving the problem in this study is correct. Moreover, the
flow is solved inside the enclosure assuming mixed convection conditions.
3.3. EVALUATION OF LAMINAR MIXED CONVECTION FLOW INSIDE A
SQUARE ENCLOSURE
In order to compare with results Basak et al. [6], Pr and Gr have been considered as 0.7 and
104, respectively. 1 < Re < 100 has also been changed.
Figures 3 and 4 Present stream function and temperature contour in comparison with
Basak et al. [6]. The acceptable harmony among figures and their results validates the
accuracy of this study.
Table 4
Ri
Some meshes used for solving the problem
Standard k-ε
RNG k-ε
RSM
Ri = 0.1 50 × 50 100 × 100 175 × 175 50 × 50 150 × 150 200 × 200 75 × 75 150 × 150 225 × 225
Ri = 1
75 × 75 125 × 125 200 × 200 75 × 75 175 × 175 250 × 250 100 × 100 175 × 175 275 × 275
Ri = 10 100 × 100 150 × 150 225 × 225 100 × 100 200 × 200 300 × 300 125 × 125 200 × 200 350 × 350
Table 5
Details of the Rectangular and Square Enclosures
Tian and Karayiannis [14], Ampofo
and Karayiannis [15], Peng and
Davidson [16], Omri and
Betts and Bokhari [19]
Galanis [17] and Hsieh
and Hsieh
and Lien [18]
and Lien [18]
Rayleigh Number
1.58 × 109
1.43 × 106
Length of Enclosure (m)
0.75
2.18
Wide of Enclosure (m)
0.75
0.076
Left Wall Temperature (°C)
50
15.6
Right Wall Temperature (°C)
10
54.7
Prandtl Number
0.707
0.697
314
Numerical modeling of turbulence mixed convection heat transfer in air filled
enclosures by finite volume method
Y = 0.1
Y = 0.2
Y = 0.3
Y = 0.4
Y = 0.5
Y = 0.6
Y = 0.7
Y = 0.8
Y = 0.9
1
0.9
Dimensionless temp
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.15
0.3
0.45
Distance from left wall (m)
0.6
0.75
Figure 1 Temperature distribution at different height, in comparison with Tian and
Karayiannis [14], Omri and Galanis [17] and Hsieh and Lien [18].
y/H = 0.05
y/H = 0.1
y/H = 0.3
y/H = 0.4
y/H = 0.6
y/H = 0.7
y/H = 0.95
y/H = 0.9
y/H = 0.5
330
325
Temprature (K)
320
315
310
305
300
295
290
285
0
0.01905
0.0381
Position (m)
0.05715
0.0762
Figure 2 Variation of temperature in Ra = 1.43 × 106, in comparison with Betts and
Bokhari [19].
3.4. EVALUATION OF TURBULENT MIXED CONVECTION FLOW INSIDE A
SQUARE ENCLOSURE
Figure 5 shows the schematics of the problem. In this case, Richardson Number varies from
0.1 to 10.
3.4.1. Validation
Figures 6 and 7 demonstrate some instances of y+ curve resulted from the present study. The
study of these curves reveals the accuracy of mesh that created in boundary layer inside the
enclosure.
Int. Jnl. of Multiphysics Volume 5 · Number 4 · 2011
315
1
6E
-05
11
00
0.0
05
6E-
0.8
1E
6E-05
0.00011
-05
0.00011
0.4
0.2
6E-
05
0.00011
0
Figure 3
0
6E-05
1E-05
0.6
0.2
0.4
0.6
0.8
1
Stream Function with Re = 10, in comparison with Basak et al. [6].
0.05
0.2
0.05
0.35
0.2
0.6
0.2
0.8
0.35
0.05
1
0.5
0.2
0.
5
65
0.
0.2
0.35
0.35
0.05
0.4
0.
5
0.8
0.65
0
Figure 4
0
0.9 0.8
0.95
8
0.2
0.4
0.6
0.8
1
Temperature Contour with Re = 1, in comparison with Basak et al. [6].
Y
Vlid
g
TH
TC
X
Figure 5
Schematic of the problem.
316
Numerical modeling of turbulence mixed convection heat transfer in air filled
enclosures by finite volume method
50
45
Top wall
Bottom wall
40
Y plus
35
30
25
20
15
10
5
0
0
Figure 6
0.1
0.2
0.3
0.4
Position (m)
0.5
0.6
0.7
0.8
y+ Diagram on Upper and Lower Walls.
70
Left wall
60
"Right wall"
Y Plus
50
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Position (m)
Figure 7 y+ Diagram on Left and Right Walls.
Figure 8 shows the vertical velocity along the med height for different turbulent and
Richardson Numbers. For Ri = 10, it s clear that the velocity on the left wall is maximum.
At boundary layer region, flow has a quick haste and thereby the velocity declines
increasingly. Upon the exit from the boundary layer, the velocity takes a more or less straight
line until the vicinity of the right wall where it reaches the cold wall boundary layer. There
again the flow takes on substantial amount of haste and hence deviates with a sharp gradient
and declines. At Ri = 0.1 where the vertical velocity has a undeniable role in the flow, The
vertical velocity is maximum on the moving wall and it declines with a pretty much even
haste as we go from the moving wall to the fixed right wall. This happens in a way that it
assumes negative values on the right hand side of the enclosure. At Ri = 1 which symbolizes
the predominance of both natural and forced convection at the same time, V behaves the
same way as in Ri = 0.1. But the gradient at this state has more tendencies towards Zero than
towards negative values. So it can be inferred that the state of V at Ri = 1 is something in
between its state at Ri = 0.1 and Ri = 10.
Int. Jnl. of Multiphysics Volume 5 · Number 4 · 2011
317
3
RNG k-e, Ri = 0.1
RNG k-e, Ri = 1
STD k-e, Ri = 0.1
STD k-e, Ri = 1
STD k-e, Ri = 10
RNG k-e, Ri = 10
Y velocity (m/s)
2.5
2
1.5
1
0.5
0
0
Figure 8
0.2
0.4
Position (m)
0.6
Vertical Velocity at y/H = 0.5.
Figure 9 shows the Nusselt Number diagram on hot wall at different Richardson
Numbers. As a result of this figure, it s clear that the Nusselt Number is maximum for the
case of Ri = 0.1 and its moderately decrease due to increase of value of Ri. According to the
determination of Richardson Number, this means that for Ri = 0.1, the forced convection
governs on fluid treatment. So the rate of heat transfer from enclosure increased as a result
of lessens Richardson Number. Moreover, the rate of heat transfer by natural convection will
be growth; if the Richardson Number will be increment. In large Richardson Numbers, the
natural convection is a major parameter of heat transfer in a enclosure. Although, this figure
demonstrates that the rate of heat transfer by mix and forced convection is much more than
natural convection.
Figures 10 to 13 are Reynolds stress graphs for RSM turbulent model and along the height
median. It is inferred from the graphs that the value of UU Reynolds stress is very close to
that of VV Reynolds stress. Also the change pattern in all UU, VV and WW Reynolds stress
graphs are similar. But the values of WW Reynolds stress are approximately 1.48 times the
value of UU and VV stresses.
14
Nusselt number
12
10
8
6
Ri = 10
Ri = 0.1
Ri = 1
4
2
0
0
0.1
0.2
0.3
0.4
Position (m)
Figure 9
Nusselt Number along the Hot Wall.
0.5
0.6
0.7
0.8
318
Numerical modeling of turbulence mixed convection heat transfer in air filled
enclosures by finite volume method
UU Reynolds stress (m2/s2)
0.15
Ri = 0.1
Ri = 1
Ri = 10
0.1
0.05
0
Figure 10
0.2
0.4
Position (m)
0.6
UU Reynolds Stress Diagram at y/H = 0.5.
VV Reynolds stress (m2/s2)
0.15
Ri = 0.1
Ri = 1
Ri = 10
0.1
0.05
0
Figure 11
0.2
0.4
Position (m)
0.6
V V Reynolds Stress Diagram at y/H = 0.5.
The UV stress assume values 10 times smaller than values of UU, VV and WW Reynolds
stresses (UV∼O (10−1)). Also UU, VV and WW stresses assume the maximum values at Ri
= 10 and minimum values at Ri = 0.1. But for the UV stress, the reverse is the case. It s
meaning that maximum values occur for forced convection governing and the minimum
values occur when free convection is governs on the flow. Also as the existing graphs and
curves for the mentioned stress show, the UU, VV and WW Reynolds stress take zero value
on the walls and take maximum value at the center of the enclosure. The UV stress has large
values on the left wall. From There after they increase up to a peak at the end of inner layer.
Then decrease almost down to x/H = 0.4 ∼ 0.47 and afterwards increase again almost up to
Int. Jnl. of Multiphysics Volume 5 · Number 4 · 2011
319
UV Reynolds stress (m2/s2)
0.25
Ri = 0.1
Ri = 1
Ri = 10
0.02
0.015
0.01
0.005
0
0
Figure 12
0.2
0.4
Position (m)
0.6
UV Reynolds Stress Diagram at y/H = 0.5.
WW Reynolds stress (m2/s2)
0.25
Ri = 0.1
Ri = 1
Ri = 10
0.2
0.15
0.1
0.05
0
Figure 13
0.2
0.4
Position (m)
0.6
WW Reynolds Stress Diagram at y/H = 0.5.
x/H∼ 0.86. There, The UV stress decline on value due to proximity to the right wall until take
negative values on the wall itself.
Figures 14 to 16 show contours of stream functions at different Richardson Numbers. It
is obvious that at Ri = 0.1, because of the strong presence of an inertia force related to the
velocity of the left wall which is surmounted on buoyancy force inside the enclosure, the entire
flow tends towards the left wall particularly the upper part where flow exits the enclosure. At
Ri = 1 the balance between the inertia and buoyancy force reduces the flow s tendency
towards the moving wall. At Ri = 10 buoyancy force has defeated the inertia force causing
the stream function to adopt a symmetrical recirculation shape. The above contour basically
320
Numerical modeling of turbulence mixed convection heat transfer in air filled
enclosures by finite volume method
0.042
0.032
0.022
0.042
04
2
2
0.012
0.6
0.
0.03
0.0
22
12
0.032
0.0
02
0.0
0.042
0.022
0.002
0.032
0.4
0.0
12
12
0.0
2
02
0.0
32
0.
0.022
0.0
42
0.032
0.2
0.032
0
0.042
0
0.2
0.4
0.6
0.8
Contour of Stream function at Ri = 0.1.
0.032
0.032
0.032
0.0
22
0.012
02
0.0
12
0.0
0.032
2
0.012
00
0.
0.022
0.012
0.4
0.032
0.6
0.022
0.022
Figure 14
0.042
0.2
0.012
0.02
2
0
Figure 15
0.022
32
0.0
0.032
0
0.2
0.4
0.6
0.8
Contour of Stream function at Ri = 1.
0.008
0.008
0.0055
0.00
03
0.0005
0.0
0.2
0.0
03
0.0
0.0055
0.008
3
0
0.0
0.0005
0.008
0.003
0.4
0.008
0.0
55
03
0.008
0.0
055
0.6
05
5
0.0055
0
Figure 16
0.008
0
0.2
0.008
0.4
Contour of Stream function at Ri = 10.
0.6
0.8
Int. Jnl. of Multiphysics Volume 5 · Number 4 · 2011
321
shows that the stream function at the center of enclosure is ten times smaller than near the
horizontal insulating wall. This implies that the flow is rather stationary at the center of the
enclosure.
Figure 17 represents turbulent kinetic energy contour at Ri = 10 for RNG k − ε Turbulent
model. Similar to the contour proposed in Omri and Galanis [17] paper, it is understood from
this contour that at free convection, the turbulence is important in the upper part of the hot
wall (left wall) and in the lower part of the cold wall (right wall). It is also inferred from this
contour that the horizontal adiabatic walls tend to the flow along the horizontal wall keep
laminar. Also these walls try to make the flow along the vertical walls and in the beginning
of the increasing region and at the end of decreasing region laminar.
As such, the turbulence increases near the hot wall between y/H = 0.2 and y/H = 0.5 and
then decreases slightly between y/H = 0.5 and y/H = 0.8 as we approach to horizontal walls.
A similar explanation is valid at the vicinity of the cold wall. The kinetic energy profile
demonstrates a perfect symmetry at Y/H = 0.5 but this symmetry does not happen at other
Y/Hs.
0.0005
0.0005
05
0.00
0.
00
05
0.0003
0.0005
0.2
0.003
0.0005
0.0005
0.4
0.0005
0.003
0.003 0.003
0.0005
0.6
0
0
Figure 17
0.2
0.4
0.6
0.8
Contour of Turbulent Kinetic Energy at Ri = 10.
4. CONCLUSIONS
In this study, Turbulence mixed convection in air filled enclosures was Modeling
Numerically by Standard k − ε, RNG k − ε and RSM turbulence models for different
Richardson Numbers. The governing equations have been solved to small reminders using
finite volume method. The results show that:
•
In large Richardson Numbers, the natural convection is a major parameter of heat
transfer in an enclosure.
•
The rate of heat transfer by mix and forced convection is much more than natural
convection.
322
Numerical modeling of turbulence mixed convection heat transfer in air filled
enclosures by finite volume method
•
At high Richardson Numbers, Turbulent kinetic energy profile demonstrates a
perfect symmetry at Y/H = 0.5 but this symmetry does not happen at other Y/Hs.
When Natural convection governs on the flow, the flow is rather stationary at the
center of the enclosure.
At all Richardson Numbers, The UV stress assumes values 10 times smaller than
values of UU, VV and WW Reynolds stresses.
For the UV stress, Maximum values occur for forced convection governing and the
minimum values occur when free convection is governs on the flow.
•
•
•
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
Bejan A., Convection heat transfer, A Wiley-Interscience publication, John Wiley, New York, 2004.
Rahman M. M., Alim M. A., Mamun M. A. H. and et al., Numerical Study of Opposing Mixed
Convection in a Vented Enclosure, ARPN Journal of Engineering and Applied Sciences, 2007,
Vol. 2, 25–35.
Saha S., M. Ali and et al., Combined Free and Forced Convection inside a Two-Dimensional
Multiple Ventilated Rectangular Enclosure, ARPN Journal of Engineering and Applied Sciences,
2006, Vol. 1, 23–35.
Saha S., Mamun A. H., Hossain M. Z. and et al., Mixed Convection in an Enclosure with
Different Inlet and Exit Configurations, Journal of Applied Fluid Mechanics, 2008, Vol. 1, 78–93.
Ghasemi B. and Aminossadati S. M., Natural Convection Heat Transfer in an Inclined Enclosure
Filled with a Water-CuO Nanofluid, Numerical Heat Transfer: Part A, 2009, Vol. 55, 807–823.
Basak T., Roy S., Sharma P. K. and et al., Analysis of Mixed Convection Flows within a Square
Cavity with Uniform and Non-Uniform Heating of Bottom Wall, International Journal of
Thermal Sciences, 2009, Vol. 48(5), 891–912.
Khanafer K.M., Al-Amiri A. M. and Pop I., Numerical Simulation of Unsteady Mixed
Convection in a Driven Cavity Using an Externally Excited Sliding Lid, European Journal of
Mechanics B: Fluids, 2007, Vol. 26, 669–687.
Goshayeshi H. R. and Safaei M. R., Investigation of turbulence mixed convection in air-filled
enclosures, Journal of Chemical Engineering and Materials Science, 2011, Vol. 2(6), 87–95.
Bessaih R. and Kadja M., Turbulent natural convection cooling of electronic components
mounted on a vertical channel, Applied Thermal Engineering, 2000, Vol. 20, 141–154.
Ampofo F., Turbulent natural convection in an air filled partitioned square cavity, International
Journal of Heat and Fluid Flow, 2004, Vol. 25, 103–114.
Salat J., Xin S., Joubert P. and et al., Experimental and numerical investigation of turbulent
natural convection in a large air-filled cavity, International Journal of Heat and Fluid Flow, 2004,
Vol. 25, 824–832.
Xaman J., Alvarez G., Lira L. and et al., Numerical study of heat transfer by laminar and turbulent
natural convection in tall cavities of facade elements, Energy and Buildings, 2005, Vol. 37, 787–794.
Aounallah M., Addad Y., Benhamadouche S. and et al., Numerical investigation of turbulent
natural convection in an inclined square cavity with a hot wavy wall, International Journal of
Heat and Mass Transfer, 2007, Vol. 50, 1683–1693.
Tian Y. S. and Karayiannis T. G., Low Turbulence Natural Convection in an Air Filled Square
Cavity, International Journal of Heat and Mass Transfer, 2000, Vol. 43, 849–866.
Ampofo, F. and Karayiannis, T. G., Experimental benchmark data for turbulent natural convection
in an air filled square cavity, International Journal of Heat and Mass Transfer, 2003, Vol. 46,
3551–3572.
Int. Jnl. of Multiphysics Volume 5 · Number 4 · 2011
323
[16] Peng S. H. and Davidson L., Large Eddy Simulation for Turbulent Buoyant Flow in a confined
cavity, International Journal of Heat and Fluid Flow, 2001, Vol. 22, 323–331.
[17] Omri M. and Galanis Ni., Numerical analysis of turbulent buoyant flows in enclosures: Influence of
grid and boundary conditions, International Journal of Thermal Sciences, 2007, Vol. 46, 727–738.
[18] Hsieh K. J. and Lien F. S., Numerical Modeling of Buoyancy-Driven Turbulent Flows
Enclosures, International Journal of Heat and Fluid Flow, 2004, Vol. 25, 659–670.
[19] Betts P. L. and Bokhari I. H., Experiments On Turbulent Natural Convection In An Enclosed Tall
Cavity, International Journal of Heat and Fluid Flow, 2000, Vol. 21, 675–683.
[20] Safaei, M. R., The Study of Laminar and Turbulence Mixed Convection Heat Transfer in
Newtonian and Non-Newtonian Fluids inside Rectangular Enclosures in Different Ri Numbers,
M. Sc. thesis, Azad University-Mashhad Branch, Iran, 2009 (in Farsi Language).
[21] Patankar S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere Washington, 1980.
[22] Goshayeshi H. R., Safaei M. R. and Maghmoomi Y., Numerical Simulation of Unsteady
Turbulent and Laminar Mixed Convection in Rectangular Enclosure with Hot upper Moving Wall
by Finite Volume Method, The 6th International Chemical Engineering Congress and Exhibition
(IChEC 2009), Kish Island, Iran, 2009.
NOMENCLATURE
u, v
Velocities in x and y Directions
x, y
Cartesian Coordinates
P
Pressure
T
Temprature
t
Time
g
Gravitational Acceleration
K
Turbulent Kinetic Energy Transport
k
Thermal Conductivity
Re
Reynolds Number
Ri
Richardson Number
Gr
Grashof Number
Nu
Nusselt Number
Pr
Prandtl Number
Ra
Rayleigh Number
ε
Dissipation of Turbulent Kinetic Energy Transport
υt
Turbulent Kinematic Viscosity
σT
Turbulent Thermal Diffusivity
β
Thermal Expansion Coefficient
ρ
Density
υ
Kinematics Viscosity
Subscripts
h
Hot Wall
c
Cold Wall
m
Mean
lid
Lid
(m/s)
(m)
(N/m2)
(K)
(sec)
(m2/s)
(m2/s2)
(W/m.k)
(m2/s3)
(m2/s)
(m2/s)
(1/K)
(kg/m3)
(m2/s)